One of the best examples of such dimensionality-control is the Ruddlesden-Popper (RP) series \(A\)\(_{n+1}\)\(B\)\(_n\)\(X\)\(_{3n+1}\) , where \(A\)  and \(B\)  are cations, \(X\)  is an anion, and \(n\)  controls the number of \(BX{_6}\)  octahedral layers. For instance, in layered ruthenates Sr\(_{n+1}\) Ru\(_n\) O\(_{3n+1}\) , the single-layer Sr\(_2\) RuO\(_4\)  (\(n\)  = 1) is an unconventional superconductor with potential \(p\) -wave pairing [1]}, [2]}, whereas the bilayer Sr\(_3\) Ru\(_2\) O\(_7\)  (\(n\)  = 2) is a paramagnetic metal with competing ferromagnetic and antiferromagnetic spin fluctuations [3]}, and it achieved a quantum critical point (QCP) by tuning the magnetic field [4]}, [5]}. In layered manganese oxide La\(_{n(1-x)}\) Sr\(_{n(x+1)}\) Mn\(_n\) O\(_{3n+1}\) , the bilayer (\(n\)  = 2) manganites exhibit colossal magnetoresistance (CMR) [6]}, but the single-layer La\(_{1+x}\) Sr\(_{1-x}\) MnO\(_4\)  does not despite a rich doping phase diagram [7]}, [8]}, [9]}, [10]}. Abrupt electronic and magnetic changes between monolayer and bilayer were also found in other families, such as lanthanum cuprates and nickelates [11]}, [12]}. While the monolayer-bilayer contrast has been highlighted in many quantum material families, what will happen if they are brought in proximity to each other to form a monolayer-bilayer double quantum well remains an interestingly open question. However, it is generally difficult to realize such a hybrid structure in real materials.
<FIGURE>